Holographic transport coefficients of dense QCD matter in paired, magnetized, and confined phases

The gauge/gravity (holographic) computation of transport coefficients for unpaired, unmagnetized, deconfined quark matter by Hoyos et al. (Phys. Rev. D 105, 066014 (2022)) deliberately covered only the simplest regime, excluding precisely the conditions neutron-star physics requires: color-superconducting quark matter at low temperature, matter in background magnetic fields with the associated anisotropic and parity-odd magnetohydrodynamic coefficients, and confined nuclear or mixed phases. The problem is to determine, within controlled holographic constructions such as V-QCD and D3-D7 probe-brane models, the full leading-order transport-coefficient matrices of dense QCD matter in these regimes. The source also raises closely related open questions that control the systematic errors and the low-temperature limits of these coefficients: the zero-temperature background structure (existence and classification of extremal solutions beyond the AdS2 family; transseries low-temperature asymptotics near the critical exponent alpha = 1/2), flavor-backreaction corrections, and whether the strong-coupling findings (Buchel-bound violation, low-temperature power-law scalings) persist in other holographic models. The core coefficients in these regimes remain uncomputed, so the problem is open.

Background

Dense QCD matter — deconfined quark matter or dense nuclear matter, at several times nuclear saturation density, thought to populate neutron-star cores and to form transiently in binary mergers — has transport properties (how it dissipates momentum, charge, and heat) that control merger hydrodynamics, magnetic-field evolution, and cooling. These properties cannot be computed reliably by perturbation theory at neutron-star densities, and lattice methods cannot access real chemical potential. The gauge/gravity duality, or holography, offers an alternative: it maps certain strongly coupled quantum field theories to classical gravity in one higher dimension, so that transport coefficients of the boundary theory are extracted from horizon data of black-hole (black-brane) solutions via Kubo formulas, which express each coefficient as the low-frequency slope of a retarded current-current correlator. Two bottom-up holographic frameworks are central here. V-QCD is a five-dimensional Einstein-dilaton-plus-tachyon model in the Veneziano limit (numbers of colors and flavors both large at fixed ratio $N_{\mathrm f}/N_{\mathrm c}$, so that flavor backreaction on the glue sector is retained), fitted to lattice and perturbative QCD data. The D3-D7 construction instead embeds flavor branes — extended objects whose low-energy dynamics is governed by the Dirac-Born-Infeld (DBI) action, a determinant-type action for brane fields — as probes in an AdS$_5\times S^5$ geometry, describing quarks at leading order without backreaction. Hoyos, Jokela, Järvinen, Subils, Tarrío and Vuorinen, building on their 2020 Letter (Phys. Rev. Lett. 125, 241601), computed the electrical and thermal conductivities and the shear and bulk viscosities $\eta$ and $\zeta$ (the coefficients of dissipative corrections to the stress tensor under shear and compression, respectively) of unpaired quark matter in beta equilibrium — the state in which weak-interaction processes have equilibrated the quark chemical potentials so the matter is electrically neutral. Their results differ dramatically from perturbative-QCD expectations, and the ratio $\zeta/\eta$, compared against the speed of sound $c_{\mathrm s}$, violates the Buchel bound, a conjectured lower bound on $\zeta/\eta$ as a function of $c_{\mathrm s}$ motivated by earlier holographic examples (Buchel, Phys. Rev. D 85, 066004 (2012)). At low temperature the V-QCD background flows to an AdS$_2\times\mathbb{R}^3$ near-horizon geometry — a two-dimensional anti-de Sitter infrared geometry signaling a quantum critical line at zero temperature and finite chemical potential (Alho, Järvinen, Kajantie, Kiritsis, Rosen, Tuominen, JHEP 04 (2014) 124) — and the coefficients approach zero-temperature asymptotics governed by transseries, i.e. expansions in simultaneous powers of $T$ and $T^{\alpha}$ with the exponent $\alpha$ fixed by the AdS$_2$ fixed point. The authors were explicit that this computation covers only the deconfined, chirally symmetric, unmagnetized, unpaired phase at leading order in flavor backreaction, and they listed what remains open: quark pairing into color-superconducting phases (CFL/2SC-type states in which quarks form Cooper pairs condensing in color-antitriplet channels; 'CFL' denotes color-flavor locking and '2SC' two-flavor color superconductivity) at low temperature, where pairing is physically expected and could qualitatively change transport; magnetohydrodynamic transport at nonzero background magnetic field $B$, where rotational symmetry is broken, many new anisotropic and parity-odd (Hall-type) coefficients appear, and their evolution couples to neutron-star thermal evolution; transport in the confined, chirally broken nuclear and mixed phases, together with holographic constructions unifying nuclear and quark-matter transport; the $N_{\mathrm f}/N_{\mathrm c}$ backreaction corrections to the probe D3-D7 results, conceptually subtle because the brane action generates a Landau pole, an ultraviolet divergence forcing the holographic dictionary to be set up at a finite cutoff surface; the existence and thermodynamic classification of strictly $T=0$ solutions at high chemical potential that appear, from near-extremal numerics, to lie away from the AdS$_2$ fixed-point family (Appendix D of the paper); analytic and numerical control of the transseries asymptotics when $\alpha$ is close to the critical value $1/2$ (for the fitted potential set 8b, $\alpha\approx0.4850$, making the competing subleading exponents $T$ and $T^{2\alpha}$ nearly degenerate); and the extension of the computation to the other models of quark matter that the authors note one might equally well employ — the Sakai-Sugimoto D8-brane model (a top-down holographic model of QCD built from D8-branes that carry the chiral flavor symmetries), Einstein-Maxwell-dilaton theories (classical gravity coupled to a Maxwell field and a single scalar, the simplest bottom-up models of dense matter), and smeared backreacted branes (flavor branes homogenized over transverse directions so that their gravitational backreaction can be retained) — so as to test whether the qualitative findings persist there.

Problem Statement

Determine, within controlled holographic (gauge/gravity) constructions of dense QCD matter, the leading-order transport coefficients in the regimes excluded by the unpaired-phase computation of Hoyos et al. (2022). Specifically, compute the hydrodynamic transport-coefficient matrices — shear and bulk viscosities, electrical and thermal conductivities, together with their anisotropic and parity-odd (Hall-type) extensions where the medium requires them — for: (1) color-superconducting (paired) quark matter at low temperature and high chemical potential; (2) unpaired quark matter at finite chemical potential in a nonzero background magnetic field, including the full set of parity-breaking coefficients needed for neutron-star magnetohydrodynamics; and (3) confined/nuclear and mixed dense phases below or across the deconfinement transition. The source raises, as closely related open questions that control the systematic errors and the $T\to0$ limits of these coefficients: (a) control of the flavor backreaction ($N_{\mathrm f}/N_{\mathrm c}$) corrections to the probe D3-D7 transport results, including the treatment of the associated Landau-pole and finite-cutoff subtleties; (b) resolution of the zero-temperature background structure at high chemical potential — whether strictly $T=0$ solutions lying away from the AdS$_2$ fixed-point family exist, their classification, and their thermodynamic dominance; (c) analytic and numerical control of the transseries low-temperature asymptotics when the exponent $\alpha$ is close to $1/2$; and (d) extension of the computation to other holographic models of quark matter (Sakai-Sugimoto, Einstein-Maxwell-dilaton, smeared backreacted branes), assessing whether the qualitative strong-coupling findings of the unpaired computation — the violation of the Buchel bound relating $\zeta/\eta$ to the speed of sound, and the low-temperature power-law scalings — persist there. The problem is to determine the transport-coefficient matrices in all three regimes (1)–(3); the closely related questions (a)–(d) bound the systematic errors and the $T\to0$ limits of these coefficients and remain open in the source.

References

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